TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 2, Pages 493–517 S 0002-9947(02)03088-X Article electronically published on October 4, 2002
THE RADIUS OF METRIC REGULARITY
A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR
Abstract. Metric regularity is a central concept in variational analysis for the study of solution mappings associated with “generalized equations”, including variational inequalities and parameterized constraint systems. Here it is em- ployed to characterize the distance to irregularity or infeasibility with respect to perturbations of the system structure. Generalizations of the Eckart-Young theorem in numerical analysis are obtained in particular.
1. Introduction Let X and Y be real Banach spaces, with norms both denoted by and closed k·k unit balls BX and BY .LetF be a mapping from X to Y , by which we will generally 1 mean a set-valued mapping, indicated by F : X ⇒ Y ,havinginverseF − : Y ⇒ X 1 with x F − (y) y F (x), and having effective graph, domain and range sets given respectively∈ ⇔ by ∈ 1 gph F = (x, y) y F (x) , dom F = x F (x) = , rge F =domF − . ∈ 6 ∅ Single-valuedness of F on a subset of X is a special case; when the subset is all of X (implying dom F = X), we say F is a single-valued mapping from X to Y and write F : X Y . This terminology→ is suited for the study of “generalized equations” of the form F (x) y and their solutions x for fixed y as parameter. When F is single-valued, such a3 relation reduces to a true equation, but more broadly it can express a mixture of inequality and equality conditions on x, interpreted as a constraint system. On the other hand, the relation F (x) y can stand for a variational inequality or a 3 1 system of optimality conditions. Either way, the set of solutions is F − (y)andis 1 nonempty if and only if y rge F . A central issue is how F − (y) behaves with respect to perturbations in ∈y as a parameter, or for that matter, perturbations in F itself. It is easy to recognize in this picture two of the abiding themes in numerical work: bounds on what happens to solutions under perturbations, and estimates of how large a perturbation can be before good behavior of a solution mapping breaks down. Our goal is to build on recent work on the first of these topics, using advanced
Received by the editors July 27, 2000. 2000 Mathematics Subject Classification. Primary 49J53; Secondary 49J52, 90C31. Key words and phrases. Metric regularity, perturbations, distance to irregularity, distance to infeasibility, Eckart-Young theorem, Lusternik-Graves theorem, Robinson-Ursescu theorem, coderivatives. Research partially supported by the NSF Grant DMS–9803098 for the first and the third author, and by the Natural Sciences and Engineering Research Council of Canada for the second author.
c 2002 American Mathematical Society
493 494 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR tools in variational analysis and convexity to reach a new level of understanding of the second topic. Far-reaching extensions of the Eckart-Young theorem on matrix perturbations will be obtained along with insights into the distance to infeasibility in the perturbation of constraint systems. 1 In our framework, where neither F nor F − needs to be single-valued, we gen- 1 erally have to focus on a particulary ¯ Y and solutionx ¯ F − (¯y), being content with local analysis around the pair (¯x,∈y¯) gph F . A key property∈ of “good behav- ior” is metric regularity. The mapping F ∈is said to be metrically regular atx ¯ fory ¯ when there exists κ (0, ) such that ∈ ∞ 1 (1.1) d(x, F − (y)) κd(y,F(x)) for all (x, y)closeto(¯x, y¯). ≤ Here, d denotes distance: thus d(y,E)=infe E y e for any set E Y .(By convention, infima over the empty set are + .)∈ Thisk − propertyk gives an estimate⊂ of how far a candidate x can be from the solution∞ set corresponding to y; the distance is bounded above by a multiple κ of the distance between y and F (x), which is a sort of “residual” that may be easy to compute. The regularity modulus of F atx ¯ fory ¯ is the value (1.2) reg F (¯x y¯):=inf κ (0, ) (1.1) holds [0, ] | ∈ ∞ ∈ ∞ (cf. Ioffe [13]). The case reg F (¯xy¯)= corresponds to the absence of any κ
(0, ) satisfying (1.1); thus, in this| notation,∞ metric regularity of F atx ¯ fory ¯ ∈is signaled∞ by reg F (¯x y¯) < . Our main result,| formulated∞ below, concerns the extent to which F can be per- turbed before metric regularity is lost. A measure of this, called the radius of metric regularity, will be introduced and characterized in terms of reg F (¯x y¯). To appreciate what is thereby achieved, it is important to understand the deep| con- nections between metric regularity and other kinds of “good behavior” of solution mappings, and to have some sense of the machinery that is already available for determining reg F (¯x y¯). | 1 Metric regularity of F is tied to a fundamental Lipschitz-type property of F − . 1 Following [25], we say F − has the Aubin property aty ¯ forx ¯ when there exists κ (0, ) together with a neighborhood O ofx ¯ such that ∈ ∞ 1 1 (1.3) F − (y0) O F − (y)+κ y0 y BX for all y,y0 close toy ¯. ∩ ⊂ k − k 1 If F − is single-valued aroundy ¯, this is ordinary Lipschitz continuity with constant 1 κ, whereas more generally, if F − is locally bounded with closed graph, it is such continuity with respect to the Pompeiu-Hausdorff metric—but (1.3) operates use- 1 fully without any boundedness requirement on the sets F − (y). A celebrated fact 1 is that F − has the Aubin property aty ¯ forx ¯ if and only if F is metrically regular atx ¯ fory ¯, and moreover the constants κ agree, so that the regularity modulus reg F (¯x, y¯) also has the formula (1.4) reg F (¯x y¯)=inf κ (0, ) (1.3) holds . | ∈ ∞ A full discussion of this relationship and its history, leading even to formulations in which X and Y are merely metric spaces, has recently been provided by Ioffe [14]; details in the finite-dimensional case, with many references beyond, are also 1 available in [25]. In the notation of [25], the right side of (1.4) is lip F − (¯y x¯)and 1 | is called the Lipschitz modulus for F − aty ¯ forx ¯. THE RADIUS OF METRIC REGULARITY 495
Another way of looking at the regularity modulus reg F (¯x y¯) is through the property of F being linearly open,orlocally surjective atx ¯ for| y ¯, which refers to the existence of κ (0, ) and a neighborhood O ofy ¯ such that ∈ ∞ (1.5) F (x+int κrBX ) F (x)+int rBY O for all x close tox ¯ and r>0. ⊃ ∩ This too is equivalent to metric regularity of F atx ¯ fory ¯ with the same range of values for κ, and thus it yields for the regularity modulus a third formula, (1.6) reg F (¯x y¯)=inf κ (0, ) (1.5) holds . | ∈ ∞ Again, details can be found in [14] and [25] and the works they cite. The meaning of metric regularity is much simpler when F is a single-valued mapping that is linear. Let L(X, Y ) denote the space of continuous linear mappings X Y ,andforF L(X, Y ), let F be the usual operator norm. → ∈ k k Example 1.1 (metric regularity of linear mappings). If F L(X, Y ), then the regularity modulus reg F (¯x y¯)isthesameforall(¯x, y¯) gph ∈F , and that common value, denoted simply by reg| F ,isgivenby ∈ (1.7) 1 reg F =inf κ (0, ) κF (BX ) int BY =sup d(0,F− (y)) y BY . ∈ ∞ ⊃ ∈ 1 Moreover, reg F< if and only if F is surjective. If in that case F − is single- valued, as must be true∞ when dim X =dimY< ,then ∞ 1 (1.8) reg F = F − . k k Thus, metric regularity is equivalent to nonsingularity when dim X =dimY< . ∞ Detail. In (1.5), the inclusion can be reduced to κF (BX ) int BY by subtracting F (x) from both sides and then dividing by r. This yields⊃ the first equation in (1.7). On the other hand, we can write the inequality for metric regularity in 1 (1.1) as d(0,F− (y) x) κd(y F (x), 0) and, by noting that linearity implies 1 1 − ≤ − 1 F − (y) x = F − (y y0)fory0 = F (x), reformulate it as d(0,F− (y y0)) − − − ≤ κ y y0 . To say that this holds for all y sufficiently close toy ¯ and y0 such that k − k 1 y0 = F (x) for an x sufficiently close tox ¯ is to say that d(0,F− (z)) κ z for all z near 0, and hence then for every z Y , again by linearity. The≤ regularityk k modulus thus is also given by the second expression∈ in (1.7). The assertion about surjectivity of F follows from (1.7) and the Banach open mapping principle (cf. [26]); the rest is then immediate. For single-valued mappings F that are nonlinear but differentiable, the notion of metric regularity, if not the term itself, goes back to a basic theorem in analysis, which is associated with the work of Lusternik [17] and Graves [11]. Here we denote by DF(¯x) the derivative mapping in L(X, Y )thatisassociatedwithF atx ¯. Theorem 1.2 (Lusternik-Graves). For any continuously Fr´echet differentiable mapping F : X Y and any (¯x, y¯) gph F , one has → ∈ (1.9) reg F (¯x y¯)=regDF(¯x). | Thus, F is metrically regular at x¯ for y¯ = F (¯x) if and only if DF(¯x) is surjective. Statements of this theorem have usually revolved around the final assertion, but the fact that reg F (¯x y¯)=regDF(¯x) can be gleaned from the proof of Graves and the observations in Example| 1.1, as transferred to the linear mapping DF(¯x). For newer work on the Lusternik-Graves theorem, see [6], [2], [3], [5], [7], [14]. 496 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR
These facts set the stage for studying perturbations and whether they can cause “irregular behavior”. The starting point is the classical Eckart-Young theorem in numerical analysis, which is usually presented in terms of n n matrices (cf. [12]) but, for the sake of enhancing comparisons, is formulated here× in terms of linear mappings. The issue is the extent to which a nonsingular linear mapping F from IRn to IRn can be perturbed by the addition of a linear mapping G without destroying the nonsingularity.
Theorem 1.3 (Eckart-Young). For all nonsingular F L(X, Y ),inthecasewhen dim X =dimY< , ∈ ∞ 1 (1.10) min G F + G singular =1/ F − . G L(X,Y ) k k k k ∈ n o
Through Example 1.1, we can identify nonsingularity in this result with metric 1 regularity, and 1/ F − with 1/ reg F . Thus, the largest radius r such that F + G is nonsingular fork all Gk L(X, Y ) satisfying G Definition 1.4 (radius of metric regularity). For any mapping F : X ⇒ Y and (¯x, y¯) gph F ,theradius of metric regularity atx ¯ fory ¯ is the value (1.11)∈ rad F (¯x y¯):= inf G F + G not metrically regular atx ¯ fory ¯ + G(¯x) . | G L(X,Y ) k k ∈ n o The value rad F (¯x y¯) could equally well be called the distance to irregularity in the sense suggested before| the definition, with respect to adding a linear mapping to F . Obviously in the Eckart-Young setting of Theorem 1.3, rad F (¯x y¯)isthesame 1 | for all (¯x, y¯) gph F , and this common value equals 1/ F − ,whichis1/ reg F by Example 1.1.∈ We aim at capturing the analogous relationshipk k locally between rad F (¯x, y¯)andregF (¯x, y¯) in much wider circumstances. Of course, for general F : X ⇒ Y there is no reason to restrict perturbations to the addition of a mapping G : X Y that belongs to L(X, Y ). An attractive feature of the radius defined by (1.11)→ is that it turns out to serve also for a vastly larger class of perturbation mappings G : X Y , at least in finite dimensions. This result is part of our main theorem, stated→ next. The theorem refers to the THE RADIUS OF METRIC REGULARITY 497 Lipschitz modulus of a single-valued mapping G at a pointx ¯,whichis G(x0) G(x) (1.12) lip G(¯x) := lim sup k − k . x,x0 x¯ x0 x x,x →=¯x 06 k − k Obviously lip G(¯x) < if and only if G is Lipschitz continuous on some neigh- borhood ofx ¯.WhenG∞is continuously differentiable, lip G(¯x)= DG(¯x) .Inthe context of the broader notion of the Lipschitz modulus of a set-valuedk mappingk in 1 [25], lip G(¯x) = lip G(¯x y¯)=regG− (¯y x¯) with respect toy ¯ = G(¯x). We also use in the theorem| statement| the concept of local closedness of a set at a point, meaning that some neighborhood of the point has closed intersection with the set. Theorem 1.5 (characterization of the radius of metric regularity). For a map- ping F : X ⇒ Y and any (¯x, y¯) gph F at which gph F is locally closed, rad F (¯x y¯) 1/ reg F (¯x y¯).Infact, ∈ | ≥ | (1.13) rad F (¯x y¯)=1/ reg F (¯x y¯) when dim X< , dim Y< . | | ∞ ∞ Furthermore, in this case, the infimum in the definition of rad F (¯x y¯) is un- changed if taken with respect to G L(X, Y ) of rank 1, but also is unchanged| when the space of perturbations G ∈is enlarged from linear mappings to locally Lipschitz continuous mappings: (1.14) rad F (¯x y¯) | =min lip G(¯x) F + G not metrically regular atx ¯ fory ¯ + G(¯x) . G:X Y → n o Thus, although defined for convenience in terms of linear perturbations, rad F (¯x y¯) is also, when X and Y are finite-dimensional, the largest radius r such that for any| G : X Y that is Lipschitz continuous aroundx ¯ with constant less than r,the mapping→ F + G is sure to be metrically regular atx ¯ fory ¯ + G(¯x). In proving Theorem 1.5, in Section 3, we actually show in more detail that for X and Y of arbitrary dimension one always has min lip G(¯x) F + G not metrically regular atx ¯ fory ¯ + G(¯x) (1.15) G:X Y → n o 1/ reg F (¯x y¯) ≥ | (see Corollary 3.4), while on the other hand, for finite-dimensional X and Y one can obtain from the calculus of coderivatives that (1.16) inf G F + G not metrically regular atx ¯ fory ¯ + G(¯x) 1/ reg F (¯x y¯). G L(X,Y ) ∈ k k ≤ | G of rank 1 n o These inequalities, and the fact that lip G(¯x)= G when G is linear, furnish the result. k k Note that it is not necessary to assume in Theorem 1.5 that F itself is metrically regular atx ¯ fory ¯, since if this were not true we would have reg F (¯x y¯)= and the formula would be correct under the convention that 1/ = 0. Likewise,| in∞ the case of reg F (¯x y¯) = 0 the formula should be interpreted as∞ giving rad F (¯x y¯)= . Groundwork| will be laid in Section 2 by studying mappings that are| positively∞ homogeneous. Graphical derivative and coderivative mappings are a prime target 498 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR among positively homogeneous mappings, for use in Section 3 in establishing the inequality in (1.16), but the results in Section 2 also are of interest in themselves. They include a different kind of generalization of the Eckart-Young theorem based on an extended definition of “nonsingularity” (see Theorem 2.6). They also estab- lish that for sublinear mappings with closed graph, the equation in Theorem 1.5 holds at the origin even when X and Y are not finite-dimensional (see Theorem 2.9). In Section 4, we apply our results to convex constraint systems, i.e., to the solu- 1 tion set F − (¯y) when gph F is convex. The question then is how far both F and y¯ can be perturbed without encountering infeasibility in the sense of the solution set becoming empty. Working with systems of certain types where F is positively homogeneous, Renegar [20] introduced a notion of the distance to infeasibility.A bridge between that notion and metric regularity is provided by the following re- sult of Robinson [21], [22], and Ursescu [27], which generalizes the Banach open mapping principle in a direction different from that taken by the Lusternik-Graves theorem. (This statement omits, as unneeded for present purposes, some accompa- nying information about the growth of the metric regularity constant κ as (¯x, y¯)is varied.) Theorem 1.6 (Robinson-Ursescu). For a mapping F : X ⇒ Y and (¯x, y¯) gph F , if F has closed convex graph, then F is metrically regular at x¯ for y¯ if and∈ only if y¯ int rge F . ∈ According to Theorem 1.6, metric regularity corresponds to “strict feasibility” of a convex constraint system—wherey ¯, at least, can be perturbed by some amount before feasibility is lost. Making use of this, we prove in Section 4 that Renegar’s distance to infeasibility is, in fact, the radius of metric regularity for a mapping F¯ derived from F andy ¯ by a process of “homogenization”. The theory of infeasibility is thereby reconfigured as a branch of the theory of metric regularity and opened up to the many resources in that wider theory, while at the same time it is extended from homogeneous to possibly inhomogeneous systems. Section 5 specializes the “generalized equation” F (x) y¯ to a variational in- equality and applies our radius theory to that important3 context, bringing out a connection with a property of strong metric regularity in which the inverse mapping 1 F − has a single-valued localization. 2. Homogeneous Mappings and Nonsingularity A mapping F : X ⇒ Y is positively homogeneous when 0 F (0) and F (λx) λF (x)forλ>0, or equivalently, when gph F is a cone in X∈ Y .Itissublinear⊃ × when, in addition, F (x + x0) F (x)+F (x0), or equivalently, when gph F is a convex cone in X Y . Sublinear⊃ mappings are also called convex processes [24] and have been the subject× of extensive development. Linear mappings are sublinear mappings in particular. Graphical derivative or coderivative mappings that are defined by way of tangent or normal cones to the graphs of other mappings are always positively homogeneous, but may or may not be sublinear, depending on 1 the circumstances. Obviously, F − is positively homogeneous if and only if F is positively homogeneous, and likewise with sublinearity. THE RADIUS OF METRIC REGULARITY 499 For a positively homogeneous mapping F : X ⇒ Y ,theouter norm and the inner norm are, respectively, + (2.1) F =sup sup y and F − =sup inf y , k k x BX y F (x) k k k k x BX y F (x) k k ∈ ∈ ∈ ∈ either of which might be .WhenF is single-valued on dom F , its outer and inner norms agree and can be∞ written as F ; this reduces to the operator norm of F when F L(X, Y ). Of course, thesek expressionsk are not true “norms” apart from the single-valued∈ case, because general set-valued mappings do not form a vector space. 1 Inner and outer norms can be applied to F − as well as to F , and that will dominate our use of them. Indeed, the expression on the right of (1.7), character- 1 izing reg F for a linear mapping F , can be identified with F − −. An analogous characterization is available for sublinear mappings F where,k however,k the focus is on reg F (0 0) instead of a regularity modulus that is the same at all points of the graph of F .| Example 2.1 (metric regularity of sublinear mappings). For a sublinear mapping F : X Y and any (¯x, y¯) gph F , ⇒ ∈ 1 (2.2) reg F (¯x y¯) reg F (0 0) = F − −. | ≤ | k k Thus, F is metrically regular everywhere when it is metrically regular at 0 for 0. In fact, (2.3) reg F (0 0) | =inf κ (0, ) F (x + κrBX ) F (x)+rBY for all x X, r > 0 . ∈ ∞ ⊃ ∈ n o Moreover, reg F (0 0) < if and only if F is surjective. | ∞ Detail. Recall from the introduction that reg F (¯x y¯) can be described as the in- fimum of all κ (0, ) for which (1.5) holds relative| to some neighborhood O of ∈ ∞ y¯. This property of κ holding for (¯x, y¯)=(0, 0) clearly implies F (κBX ) int BY ; intersection with a neighborhood O is superfluous because of positive homogene-⊃ ity. On the other hand, just from knowing that F (κBX ) int BY for a certain κ>0, we obtain for arbitrary (x, y) gph F and r>0 through⊃ the sublinearity ∈ of F that F (x + κrBX ) F (x)+rF(κBX ) y +intrBY . This establishes that reg F (¯x y¯) reg F (0 0)⊃ for all (¯x, y¯) gph⊃F and, appealing again to positive homogeneity,| ≤ that | ∈ (2.4) reg F (0 0) = inf κ (0, ) F (κBX ) BY . | ∈ ∞ ⊃ 1 1 By definition, F − − =inf κ (0, ) y B Y F − (y) κBX = . Hence 1 k k ∈ ∞ ∈ ⇒ ∩ 6 ∅ F − equals the right side of (2.4) as well. The fact that the finiteness of the − kvaluek in (2.4) corresponds to F being surjective is an immediate consequence of the Robinson-Ursescu result in Theorem 1.6, but it was actually proved earlier by Robinson in [21]. Next, as a basis for generalizing the Eckart-Young theorem, we use outer norms of inverses to extend the definition of nonsingularity from linear mappings X Y with dim X =dimY< to general positively homogeneous mappings X → Y ∞ ⇒ 500 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR without any dimensionality restrictions. Later, when we return to metric regular- ity, such nonsingularity will be important as a dual property applied to adjoint mappings or coderivative mappings. Definition 2.2 (extended nonsingularity). A positively homogeneous mapping F : 1 + 1 + X Y will be called singular if F − = , nonsingular if F − < . ⇒ k k ∞ k k ∞ For linear F : X Y , nonsingularity in this sense coincides with the traditional → 1 notion when dim X =dimY< , but it means in general that F − is single- valued and continuous relative to∞ rge F , its domain. For positively homogeneous 1 mappings that are not linear, nonsingularity implies F − (0) = 0 but does not necessitate single-valuedness elsewhere. { } Example 2.3 (extended nonsingularity in finite dimensions). When dim X< , ∞ dim Y< , a positively homogeneous F : X ⇒ Y with closed graph is nonsingular ∞ 1 if and only if F − (0) = 0 . { } Example 2.4 (extended nonsingularity of sublinear mappings). A mapping F : 1 X ⇒ Y that is sublinear is both nonsingular and surjective if and only if F = L− for a continuous linear mapping L : Y X. → 1 Detail. The assumptions imply that F − is a sublinear mapping L with dom L = Y and L(0) containing only 0. Then for any y Y we have both L(y)andL( y) nonempty with L(0) = L(y y) L(y)+L( ∈y), so that L must be single-valued− everywhere with L( y)=L−(y).⊃ In combination− with positive homogeneity, this yields L(λy)=λL(−y) for all real λ. Single-valuedness in the sublinearity rule L(y + y0) L(y)+L(y0)requiresL(y + y0)=L(y)+L(y0), and so L must be linear. ⊃ 1 + Furthermore, L = F − < . k k k k ∞ 1 + The following characterization of F − will help to elucidate the meaning of extended nonsingularity more generally.k k Proposition 2.5 (inverse norm formula). For a positively homogeneous mapping F : X ⇒ Y , (2.5) 1 + 1 1 F − =inf κ (0, ) x F − (y) x κ y =sup . k k ∈ ∞ ∈ ⇒kk≤ k k x =1 d(0,F(x)) k k 1 Proof. By the definition of the outer norm in (2.1) as applied to F − , we know that 1 + 1 F − is the supremum of x over all pairs (y,x) gph F − with y 1. Thus, k k k k 1 ∈ k k≤ it is the infimum of all κ (0, ) such that F − (BY ) κBX. That translates through positive homogeneity∈ to∞ the middle expression in⊂ (2.5). There we observe that the infimum is unchanged when x is restricted to have x =1,inwhich case the expression can be identified with the infimum of all κk k(0, ) such that κ 1/ y whenever y F (x)and x = 1. (It is correct in∈ this∞ to interpret 1/≥y =k k if y = 0.) This∈ shows thatk thek middle expression in (2.5) agrees with k k ∞ the final expression in (2.5). Theorem 2.6 (extended Eckart-Young). For any F : X ⇒ Y that is positively homogeneous, 1 + (2.6) inf G F + G singular =1/ F − . G L(X,Y ) k k k k ∈ n o THE RADIUS OF METRIC REGULARITY 501 Moreover, the infimum is the same if restricted to mappings G L(X, Y ) of rank ∈ one. (Indeed, if X = Z∗ for a Banach space Z, the additional restriction can be made that G is weak*-to-norm continuous.) Proof. If F is singular, (2.6) holds with 0 on both sides; so we can assume that 1 + 1 + 1 + F − < . We can also assume that F − > 0, because having F − =0 correspondsk k ∞ to having dom F = 0 , andk that impliesk for any G that dom(k F +k G)= 1 + { } 0 ; hence (F + G)− =0.Inthatcase,(2.6)holdswith on both sides. { }SupposekG L(X,k Y )andF + G is singular. Then by∞ definition there is a ∈ sequence of elements (xk,yk) gph(F +G)with yk 1and0< xk .From ∈ 1 k k≤ k 1 k→∞+ yk (F +G)(xk)wehavexk F − (yk G(xk)); hence xk F − yk G(xk) and∈ consequently, ∈ − k k≤k k k − k 1 + 1/ F − ( yk + G(xk) )/ xk (1/ xk )+ G . k k ≤ k k k k k k≤ k k k k 1 + Taking the limit as k ,weget1/ F − G .Thus“ ” holds in (2.6). →∞ k k ≤k k ≥ 1 + We have to show now that “ ” holds as well. Consider any finite r>1/ F − . 1 + ≤ k k We have 1/r < F − ;sotheremustexist(ˆx, yˆ) gph F with yˆ =1and k k ∈ k k xˆ > 1/r. In the dual space X∗ it is possible to findx ˆ∗ with x,ˆ xˆ∗ = xˆ and k k ˆ ˆ h i 1 k k xˆ∗ = 1. Define the rank-one mapping G L(X, Y )byG(x)= xˆ − x, xˆ∗ yˆ. k k ∈ −k k h i Then Gˆ(ˆx)= yˆ,sothat(F + Gˆ)(ˆx)=F (ˆx)+Gˆ(ˆx)=F (ˆx) yˆ 0. Therefore ˆ −1 ˆ − 3 ˆ xˆ (F + G)− (0), and F + G must be singular. On the other hand, G = yˆ∈/ xˆ =1/ xˆ (y∗,x∗) gph F ∗− x∗,x y∗,y for all (x, y) gph F. ∈ ⇔h i≥h i ∈ + The graphs of both F ∗ and F ∗− correspond to the closed convex cone in X∗ Y ∗ × that is polar to gph F , except for permuting (x∗,y∗)to(y∗,x∗) and introducing + certain changes of sign. In particular, both F ∗ and F ∗− are always sublinear with + closed graph; furthermore, gph F ∗− = gph F ∗ . The interesting fact is that when F itself is sublinear with closed graph,− one has (see [1] or [25, 11.29]): + + (2.8) (F ∗ )∗− =(F ∗−)∗ = F, and moreover, + + + + + (2.9) F = F ∗ − = F ∗− − and F − = F ∗ = F ∗− . k k k k k k k k k k k k 502 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR Proposition 2.7 (adjoint formula for metric regularity). For a sublinear mapping F : X ⇒ Y with closed graph, the characterizations of reg F (0 0) in Example 2.1 are supplemented by | + 1 + (2.10) reg F (0 0) = (F ∗ )− . | k k 1 1 Proof. We know from Example 2.1 that reg F (0 0) = F − −. However, F − − = 1 + 1 | k k 1 k k+ 1 (F − )∗− by (2.9) (as applied to F − instead of F ). Also, (F − )∗− =(F ∗ )− k k by the adjoint formulas. This leads to (2.10). The relations between adjoints and their norms also induce a duality between the properties of surjectivity and nonsingularity. Proposition 2.8 (Borwein [1]). For sublinear F : X ⇒ Y with closed graph, F is + surjective if and only if F ∗ is nonsingular. Likewise, F is nonsingular if and only + if F ∗ is surjective. Proof. This is immediate from Example 2.1, Definition 2.2 and the relations in (2.9). Theorem 2.9 (radius of metric regularity for sublinear mappings). For a sublinear mapping F : X ⇒ Y with closed graph, (2.11) rad F (0 0) = 1/ reg F (0 0) = inf G F + G not surjective . | | G L(X,Y ) k k ∈ n o Moreover, the infimum is the same if restricted to G of rank one. Proof. For any G L(X, Y ), the mapping F + G is sublinear with closed graph + ∈ + and has (F + G)∗ = F ∗ + G∗. Thus by Proposition 2.8, F + G is surjective if and + only if F ∗ + G∗ is nonsingular. It follows that (2.12) + inf G F + G not surjective =inf G∗ F ∗ + G∗ singular . G L(X,Y ) k k G L(X,Y ) k k ∈ n o ∈ n o The right side of (2.12) can be identified through Theorem 2.6 with + + 1 + (2.13) inf H F ∗ + H singular =1/ (F ∗ )− H L(Y ∗,X∗) k k k k ∈ n o by the observation that any H L(Y ∗,X∗) of rank one that is weak*-to-norm ∈ continuous has the form G∗ for some G L(X, Y ) of rank one. Consequently, the left side of (2.12) equals the right side of∈ (2.13), and from (2.10) we then get the equality between the second and third expressions in (2.11). The third expression in (2.11) equals the first by Definition 1.4 and the assertion at the end of Example 2.1. 1 Note that since reg F (0 0) = F − − (cf. Example 2.1), the second part of (2.12) can be written equivalently| k as k 1 inf G F + G not surjective =1/ F − −. G L(X,Y ) k k k k ∈ n o This relation was recently established by Lewis [15, Thm. 4.8], but here it has been deduced from broader facts about generalized nonsingularity, as well as placed in a context of metric regularity. In line with the observation after Theorem 2.6, the infimum in (2.11) would be unaffected if taken over sublinear G : X ⇒ Y such that dom G = X and F + G THE RADIUS OF METRIC REGULARITY 503 has closed graph, as long as G is replaced by G +. The proof still works, with + k k k k G∗ replaced by G∗ . Remark. Once we have established Theorem 1.5, it will be possible to regard The- orem 2.9 as a special case of that result (by way of Theorem 1.6) when X and Y are finite-dimensional, and then the additional information in Theorem 1.5 about nonlinear perturbations could be brought in. Theorem 1.5 does not cover Theorem 2.9 for infinite-dimensional X or Y , however. Corollary 2.10 (adjoint formula for the radius). For any sublinear mapping F : X ⇒ Y with closed graph, + (2.14) rad F (0 0) = inf d(0,F∗ (y∗)). | y =1 k ∗k Proof. The left side of (2.14) is 1/ reg F (0 0) by Theorem 2.9, but the right side + 1 + | is 1/ (F ∗ )− by Proposition 2.5. The two sides are therefore equal by the k k observation in Proposition 2.7. 3. Coderivatives and the Radius Characterization Graphical differentiation of set-valued mappings provides a powerful tool for dealing with metric regularity. Our use of this tool will be limited to a finite- dimensional context, for reasons to be explained, and we therefore tailor the de- scription of graphical differentiation to such spaces as well. Details beyond what we say here can be found in [25]. Let dim X< and dim Y< , and consider a mapping F : X ⇒ Y .The graphical derivative∞ of F atx ¯ for an∞ elementy ¯ F (¯x) is the mapping DF(¯x y¯) having as its graph the tangent cone to gph F at∈ (¯x, y¯): | 1 gph DF(¯x y¯) = lim sup [gph F (¯x, y¯) . | λ 0 λ − & Thus, DF(¯x y¯) is a positively homogeneous mapping X ⇒ Y with closed graph. On the basis| of the definitions reviewed in the preceding section, it has an upper + adjoint DF(¯x y¯)∗ : Y ∗ ⇒ X∗ and also a lower adjoint. The coderivative of F at x¯ fory ¯ is defined| by taking limits of the graphs of the upper adjoints at nearby points (x, y): + gph D∗F (¯x y¯) = lim sup gph DF(x y)∗ , | (x,y) (¯x,y¯) | → where (x, y) gph F . Therefore, D∗F (¯x y¯) is a positively homogeneous mapping ∈ | Y ∗ ⇒ X∗ with closed graph. The value of coderivative mappings comes from the extensive calculus that is available for determining them from the structure of a given mapping F (cf. Chap. 10 of [25]) and from the following result of Mordukhovich; for the somewhat com- plicated story of this result, in which Ioffe also had a role, see p. 418 of [25]. Theorem 3.1 (Mordukhovich criterion). For F : X ⇒ Y and any (¯x, y¯) gph F at which gph F is locally closed, if dim X< and dim Y< ,then ∈ ∞ ∞ 1 + (3.1) reg F (¯x y¯)= D∗F (¯x y¯)− . | k | k This can be translated through our Definition 2.2 into an equivalence between metric regularity and coderivative nonsingularity. 504 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR Corollary 3.2 (coderivative nonsingularity). Under the theorem’s assumptions, F is metrically regular at x¯ for y¯ if and only if the coderivative mapping D∗F (¯x y¯) is nonsingular. | Coderivative mappings can be defined in a similar manner when X and Y are infinite-dimensional, but the question of exactly what kind of set limit to take is more subtle and can depend on the “smoothness” of these spaces. Anyway, formula (3.1) can fail in the more general setting. See Mordukhovich [18] for more on this matter, including certain substitutes for (3.1) in terms of limits. The validity of (3.1) itself will be crucial to our argument for establishing equation (1.3) of Theorem 1.5, and is the reason why that equation is asserted only for finite-dimensional spaces. Another ingredient of our proof of Theorem 1.5 will be the following estimate. This estimate can be deduced in various ways from the literature on the Lusternik- Graves theorem, such as Dmitruk, Milyutin and Osmolovski˘ı [5]; analogous devel- opments appear in Theorem 1.4 of Dontchev [6] or Section 1 of the more recent paper by Ioffe [14], for example. We express the estimate here in the notation of reg F and lip G and, for completeness, supply a direct proof which is in line with the original arguments of Lusternik and Graves. Theorem 3.3 (estimate for Lipschitz perturbations). Consider any mapping F : X ⇒ Y and any (¯x, y¯) gph F at which gph F is locally closed. Consider also a ∈ 1 mapping G : X Y .Ifreg F (¯x y¯) <κ< and lip G(¯x) <λ<κ− ,then → | ∞ 1 1 κ (3.2) reg(F + G)(¯x y¯ + G(¯x)) < (κ− λ)− = . | − 1 λκ − Proof. With the notation Ba(¯x)=¯x + aBX and Ba(¯y)=¯y + aBY ,leta>0be small enough that gph F is closed relative to Ba(¯x) Ba(¯y), G is Lipschitz with × constant λ on Ba(¯x), and 1 d(x, F − (y)) κd(y,F(x)) for all (x, y) Ba(¯x) Ba(¯y). ≤ ∈ × 1 In particular, this implies that d(¯x, F − (y)) κd(y,F(¯x)) κ y y¯ when y ≤ ≤ k − k ∈ Ba(¯y), and therefore 1 (3.3) F − (y) = for all y Ba(¯y). 6 ∅ ∈ Choose α such that (3.4) 0 <α< 1 a(1 κλ)min 1,κ . 4 − { } Let x B (¯x)andy B (¯y). Then, from the choice of α in (3.4), we have ∈ α/4 ∈ α/(4κ) (3.5) y G(x)+G(¯x) y¯ λ x x¯ + y y¯ (λα/4) + (α/4κ) a. k − − k≤ k − k k − k≤ ≤ Fix ε such that 1 (3.6) 0 <ε< a(1 κλ)min 1, 1/λ . 4 − { } 1 Then, through (3.3) and (3.5), there exists z1 F − (y G(x)+G(¯x)) such that ∈ − 1 (3.7) z1 x d(x, F − (y G(x)+G(¯x))) + ε. k − k≤ − We obtain from metric regularity that 1 z1 x x x¯ + d(¯x, F − (y G(x)+G(¯x))) + ε k − k≤k− k − x x¯ + κd(y G(x)+G(¯x),F(¯x)) + ε ≤k− k − x x¯ + κ y y¯ + κλ x x¯ + ε ≤k− k k − k k − k (α/4) + (κα/4κ)+(κλα/4) + ε (3α/4) + ε. ≤ ≤ THE RADIUS OF METRIC REGULARITY 505 Then we have z1 x¯ z1 x + x x¯ (3α/4) + ε +(α/4) α + ε a. k − k≤k − k k − k≤ ≤ ≤ By induction now, we construct an infinite sequence of vectors zj, j =1, 2,...,such that 1 j zj+1 F − (y G(zj)+G(¯x)) and zj+1 zj (κλ) z1 x . ∈ − k − k≤ k − k Suppose that we have generated such z2,...,zn from z1.Thenforj =1, 2,...,n 1 we have − j 1 j 1 − − i zj x¯ zi+1 zi + z1 x¯ (κλ) z1 x + z1 x¯ k − k≤ k − k k − k≤ k − k k − k i=1 i=0 X X 1 1 z1 x + z1 x¯ (3α/4+ε)+α + ε a ≤ 1 κλk − k k − k≤ 1 κλ ≤ − − according to the choice of the constants α in (3.4) and ε in (3.6). Also, y G(zj )+G(¯x) y¯ y y¯ + λ zj x¯ k − − k≤k − k k − k α λ + (3α/4+ε)+λ(α + ε) a. ≤ 4κ 1 κλ ≤ − 1 The metric regularity of F yields the existence of zn+1 F − (y G(zn)+G(¯x)) such that ∈ − zn+1 zn κd(y G(zn)+G(¯x),F(zn)). k − k≤ − 1 Since zn F − (y G(zn 1)+G(¯x)), this implies ∈ − − zn+1 zn κ G(zn) G(zn 1) κλ zn zn 1 , k − k≤ k − − k≤ k − − k and the induction step is complete. The sequence zn satisfies the Cauchy condition, hence is convergent to some z, 1 which, from the local closedness of gph F ,satisfiesz F − (y G(z)+G(¯x)), that 1 ∈ − is, z (F + G)− (y + G(¯x)). Moreover, ∈ n 1 d(x, (F + G)− (y + G(¯x)) z x lim zi+1 zi + z1 x ≤k − k≤n k − k k − k →∞ i=1 n X i 1 lim (κλ) z1 x z1 x ≤ n k − k≤1 κλk − k →∞ i=0 X − κ d(y + G(¯x), (F + G)(x)) + ε , ≤ 1 κλ − the final inequality being obtained from (3.7). Since the left side does not depend on ε, which can be arbitrary small, this tells us that F + G is metrically regular at x¯ fory ¯ + G(¯x). Corollary 3.4 (general perturbation inequality). If the mapping F : X ⇒ Y is locally closed at (¯x, y¯) gph F ,then ∈ inf lip G(¯x) F + G not metrically regular at x¯ for y¯ + G(¯x) (3.8) G:X Y → n o 1/ reg F (¯x y¯). ≥ | Proof. If lip G(¯x) < 1/ reg F (¯x y¯), there exist λ>lip G(¯x)andκ>reg F (¯x y¯) such that λ<1/κ,andthenF |+ G must be metrically regular atx ¯ fory ¯ + G(¯|x) by the theorem. 506 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR Proof of Theorem 1.5. It is obvious from the definition of rad F (¯x y¯) in (1.11) that | (3.9) rad F (¯x y¯) | inf lip G(¯x) F + G not metrically regular atx ¯ fory ¯ + G(¯x) . ≥ G:X Y → n o The double inequality obtained by combining (3.8) and (3.9) holds without any restriction on the dimensions of X and Y . It reduces our task to demonstrating that rad F (¯x y¯)=1/ reg F (¯x y¯)whendimX< and dim Y< ,alongwith verifying that| the infimum in the| definition of rad∞F (¯x y¯) also is unchanged∞ when restricted to mappings G L(X, Y )ofrankone. | In this finite-dimensional∈ setting, we have Theorem 3.1 and Corollary 3.2 at our disposal. With these results we can restate the targeted equation as 1 + (3.10) inf G D∗(F + G)(¯x y¯ + G(¯x)) singular =1/ D∗F (¯x y¯)− . G L(X,Y ) k k | k | k ∈ n o It is elementary from the calculus of coderivatives (cf. 10.43 of [25]) that D∗(F + G)(¯x y¯ + G(¯x)) = D∗F (¯x y¯)+G∗. | | Also, G∗ = G < . Hence (3.10) can be identified with k k k k ∞ 1 + (3.11) inf G∗ D∗F (¯x y¯)+G∗ singular =1/ D∗F (¯x y¯)− . G L(X,Y ) k k | k | k ∈ n o Furthermore, again by finite-dimensionality, every H L(Y ,X )istheadjointG ∗ ∗ ∗ of some G L(X, Y ). Therefore (3.11) is valid, being∈ the special case of Theorem ∈ 2.6 as applied to the mapping D∗F (¯x y¯):Y ∗ ⇒ X∗, and on the basis of that theorem, the infimum is unchanged if taken| over G L(X, Y )ofrankone. ∈ Theorem 1.5 also gives information on what happens to the radius of metric regularity under perturbations. Corollary 3.5 (perturbed radius of metric regularity). For any F : X ⇒ Y with dim X< and dim Y< ,andany(¯x, y¯) gph F at which gph F is locally closed, one∞ has ∞ ∈ (3.12) rad(F + G)(¯x y¯ + G(¯x)) rad F (¯x y¯) lip G(¯x) when lip G(¯x) < rad F (¯x y¯). | ≥ | − | Proof. For such F and G, suppose H : X Y has lip H(¯x) < rad F (¯x y¯) lip G(¯x). Is (F + G)+H sure to be metrically regular→ atx ¯ for (¯y + G(¯x)) +| H−(¯x)? Yes, because in terms of G0 = G+ H,thisisthesameasF + G0 being metrically regular atx ¯ fory ¯ + G0(¯x), and that is true by (1.14) of Theorem 1.5 since lip G0(¯x) ≤ lip G(¯x) + lip H(¯x) < rad F (¯x, y¯). A further conclusion can be drawn. Recall that a mapping F 0 : X ⇒ Y is said to give a first-order approximation to a mapping F : X ⇒ Y around (¯x, y¯) gph F if on some neighborhood O ofx ¯, there is a mapping G : O Y with G(¯∈x)=0, → lip G(¯x) = 0, and F 0 = F + G. Corollary 3.6 (radius stability under first-order approximation). When dim X< and dim Y< ,ifF : X Y has graph locally closed at (¯x, y¯) gph F ,and ∞ ∞ ⇒ ∈ Φ:X ⇒ Y is any mapping that furnishes a first-order approximation to F around (¯x, y¯),then (3.13) rad F (¯x y¯)=radΦ(¯x y¯). | | THE RADIUS OF METRIC REGULARITY 507 Proof. Consider G : O Y as in the preceding definition of first-order approx- imation, and extend G →in any way to a mapping X Y . Since Φ agrees with F + G aroundx ¯,andG(¯x)=0,wehaveradΦ(¯x y¯)=rad(→ F + G)(¯x y¯). On the other hand, since lip G(¯x)=0wehaverad(F +G)(¯|x y¯) rad F (¯x y¯) by| Corollary 3.5. Therefore rad Φ(¯x y¯) rad F (¯x y¯). To say that| Φ≥ gives a first-order| approx- imation of F is to say| equally,≥ however,| that F gives a first-order approximation of Φ; the relationship is symmetric, with G replacing G. Hence the inequality rad F (¯x y¯) rad Φ(¯x y¯)mustholdaswell.− | ≥ | An example of a first-order approximation to which Corollary 3.6 can be applied is seen when F (x)=F0(x)+f(x) for a mapping f : X Y that is strictly → differentiable atx ¯,andΦ(x)=F0(x)+g(x)forg(x)=f(¯x)+ f(¯x),x x¯ . In this case, Φ = F + G for a mapping G that is strictly differentiableh∇ atx ¯−withi G(¯x) = 0 and Jacobian G(¯x) = 0 and thus has lip G(¯x)=0. ∇ 4. Application to Constraint Systems In this section, fully back in the context of possibly infinite-dimensional spaces X and Y , we interpret the “generalized equation” F (x) y¯ as a constraint system on x, focusing on the case where F has convex graph.3 Such a system is called 1 feasible if F − (¯y) = , i.e.,y ¯ rge F ,andstrictly feasible if actuallyy ¯ int rge F . Recall that any6 closed,∅ convex∈ cone K Y induces a partial ordering∈ “ ” ⊂ ≤K under the rule that y0 y1 means y1 y0 K. Correspondingly, y0 < y1 ≤K − ∈ K means y1 y0 int K. − ∈ Example 4.1 (convex constraint systems). Let C X be a closed convex set, let K Y be a closed convex cone, and let A : C Y⊂be a continuous mapping that is convex⊂ with respect to the partial ordering in→Y induced by K: A((1 θ)x0 + θx1) (1 θ)A(x0)+θA(x1)forx0,x1 C when 0 <θ<1. − ≤K − ∈ Define the mapping F : X ⇒ Y by A(x)+K if x C, F (x)= ∈ if x/C. (∅ ∈ Then F has closed, convex graph, and feasibility of the system F (x) y¯ refers to 3 x¯ C such that A(¯x) y,¯ ∃ ∈ ≤K while, as long as int K = , strict feasibility refers to 6 ∅ x¯ C such that A(¯x) < y.¯ ∃ ∈ K Detail. Only the final claim needs a comment. When int K = ,wehaveK = cl int K, so that the convex set rge F = A(C)+K is the closure6 ∅ of the open set O := A(C)+intK.Also,O is convex. If follows then that int rge F = O. Example 4.2 (linear-conic constraint systems). Add to Example 4.1 the assump- tion that A is linear and C is a cone, so that the conditionx ¯ C can be written ∈ + equivalently asx ¯ C 0. Then F is sublinear. Furthermore, its adjoint F ∗ : Y ∗ ⇒ ≥ + X∗ is given in terms of the adjoint A∗ of A and the dual cones K = K∗ and + − C = C∗ (where ∗ denotes polar) by − + + + A∗(y∗) C if y K , F ∗ (y∗)= − ∈ if y/K +, (∅ ∈ 508 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR + so that F ∗ (y∗) x∗ if and only if y∗ + 0andA∗(y∗) + x∗. 3 ≥K ≥C In general, for the system F (x) y¯, we will be interested in perturbations in which F is replaced by F + G andy ¯3byy ¯ + g with G L(X, Y )andg Y . Such a perturbation, the magnitude of which is quantified with∈ ∈ (4.1) (G, g) =max G , g , k k k k k k transforms the condition F (x) y¯ to (F+ G)(x) y¯ + g and the solution set 1 1 3 3 1 F − (¯y)to(F + G)− (¯y + g), creating infeasibility if (F + G)− (¯y + g)= , i.e., if y¯ + g/rge(F + G). ∅ Such∈ perturbations in the linear-conic context of Example 4.2 were studied by Renegar [20]. For that case, he introduced a measure of how large (G, g)canbe before the system (F + G)(x) y¯ + g becomes infeasible, and a sizable literature has since grown around this; see3 e.g. [10], [9], [19]. We extend Renegar’s definition to general convex systems as follows. Definition 4.3 (distance to infeasibility). For F : X ⇒ Y with convex graph and y¯ rge F ,thedistance to infeasibility of the system F (x) y¯ is the value ∈ 3 (4.2) inf (G, g) y¯ + g/rge(F + G) . G L(X,Y ),g Y k k ∈ ∈ ∈ n o It will be essential to our approach in analyzing this value that there would be no difference if feasibility were replaced by strict feasibility in the definition. Lemma 4.4 (infeasibility versus strict infeasibility). The distance to infeasibility in the sense of Definition 4.3 is the same as the distance to strict infeasibility, namely the value (4.3) inf (G, g) y¯ + g/int rge(F + G) . G L(X,Y ),g Y k k ∈ ∈ ∈ n o Proof. Let S1 denote the set of (G, g) over which the infimum is taken in (4.2) and let S2 be the corresponding set in (4.3). Obviously S1 S2; so the first infimum cannot be less than the second. We must show that it also⊂ cannot be greater. This amounts to demonstrating that for any (G, g) S2 and any ε>0, we can find ∈ (G0,g0) S1 such that (G0,g0) (G, g) + ε. In fact, we can get this with ∈ k k≤k k G0 = G simply by noting that wheny ¯ + g/int rge(F + G)theremustexistg0 Y ∈ ∈ withy ¯ + g0 / rge(F + G)and g0 g ε. ∈ k − k≤ Corollary 4.5 (reduction to metric regularity). The distance to infeasibility of the system F (x) y¯, as in Definition 4.3, is also the value 3 (4.4) inf (g,G) F +G not metrically regular at any x¯ for y¯ + g . (G,g) L(X,Y ) Y k k ∈ × n o Proof. This is now immediate from Theorem 1.6 (Robinson-Ursescu) and Definition 1.4 of the radius of metric regularity. Although Corollary 4.5 expresses the distance to infeasibility in terms of metric regularity, it does not tie in with our theory of the radius of metric regularity. For that, we have to pass from F to a special mapping F¯ constructed as a “homoge- nization” of the convex system F (x) y¯. We will then be able to apply to F¯ the result on distance to metric regularity3 in Theorem 2.9. Following[25],wedenotebyF ∞ the horizon mapping associated with F ,the graph of F ∞ in X Y being the horizon cone of the graph of F .WhenF has × THE RADIUS OF METRIC REGULARITY 509 closed, convex graph, this is the same as gph F ∞ being the recession cone of gph F in the sense of convex analysis: (x0,y0) gph F ∞ gph F +(x0,y0) gph F. ∈ ⇔ ⊂ Definition 4.6 (homogenization). For F : X ⇒ Y andy ¯ rge F ,thehomog- enization of the constraint system F (x) y¯ is the system∈F¯(x, t) 0, where F¯ : X IR Y is defined by 3 3 × ⇒ 1 tF (t− x) ty¯ if t>0, ¯ − (4.5) F (x, t)=F ∞(x)ift =0, if t<0, ∅ and the solution sets to the two systems are related by 1 1 (4.6) x F − (¯y) (x, 1) F¯− (0). ∈ ⇔ ∈ 1 Note that if F is positively homogeneous with closed graph, then tF (t− x)= F (x)=F ∞(x) for all t>0, so that we simply have F¯(x, t)=F (x) ty¯ for t 0 but F¯(x, t)= for t<0. − ≥ In what follows,∅ we adopt the norm (4.7) (x, t) = x + t for (x, t) X IR. k k k k | | ∈ × The case of the next theorem in which the mapping F is sublinear reduces to the main result of Lewis [16, Thm. 4.5], and further specialization to Example 4.2 then captures Renegar’s result [20, Thm. 1.3]. Theorem 4.7 (radius characterization of the distance to infeasibility). Let F : X ⇒ Y have closed, convex graph and let y¯ rge F . Then in the homogenized system F¯(t, x) 0, the mapping F¯ is sublinear with∈ closed graph, and 3 (4.8)y ¯ int rge F 0 int rge F¯ F¯ is surjective. ∈ ⇔ ∈ ⇔ Furthermore, for the given constraint system F (x) y¯, one has 3 (4.9) distance to infeasibility =radF¯(0, 0 0) = 1/ reg F¯(0, 0 0). | | Proof. The definition of F¯ corresponds to gph F¯ being the closed convex cone in X IR Y that is generated by (x, 1,y y¯) (x, y) gph F . Hence F¯ is sublinear, × × − ∈ and also, rge F¯ is a convex cone. We have (rge F ) y¯ = F (X) y¯ = F¯(X, 1). So it is obvious that ify ¯ int rge F ,then0 int rge F¯−.SincergeF¯−is a convex cone, the latter is equivalent∈ to having rge F¯ =∈Y , i.e., surjectivity. Conversely now, suppose F¯ is surjective; Theorem 1.6 (Robinson-Ursescu) in- formsusthatinthiscase,0 int F¯(W ) for every neighborhood W of the ori- gin in IR X. It must be verified,∈ however, thaty ¯ int rge F .Intermsof × ∈ C(t)=F¯(BX ,t) Y , it will suffice to show that 0 int C(t)forsomet>0. Note that the sublinearity⊂ of F¯ implies that ∈ (4.10) C((1 θ)t0 + θt1) (1 θ)C(t0)+θC(t1)for0<θ<1. − ⊃ − 1 Our assumption thaty ¯ rge F ensures that F − (¯y) = .Chooseτ (0, )small ∈ 1 6 ∅ ∈ ∞ enough that 1/(2τ) >d(0,F− (¯y)). Then (4.11) 0 C(t) for all t [0, 2τ], ∈ ∈ 510 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR whereas, because [ 2τ,2τ] BX is a neighborhood W of the origin in IR X,we have − × × (4.12) 0 int F¯(BX , [ 2τ,2τ]) = int C(t). ∈ − 0 t 2τ ≤[≤ We will use this to show that actually 0 int C(τ). For y∗ Y ∗ define ∈ ∈ σ(y∗,t):= sup y,y∗ ,λ(t):= inf σ(y∗,t). y C(t)h i y∗ =1 ∈ k k The property in (4.10) makes σ(y∗,t)concaveint, and the same then follows for λ(t). As long as 0 t 2τ,wehaveσ(y∗,t) 0andλ(t) 0 by (4.11). On the other hand, the union≤ in≤ (4.12) includes some≥ ball around the≥ origin. Therefore, (4.13) ε>0 such that sup σ(y∗,t) ε for all y∗ Y ∗ with y∗ =1. ∃ 0 t 2τ ≥ ∈ k k ≤ ≤ We argue next that λ(τ) > 0. If not, then since λ is a nonnegative concave function on [0, 2τ], we would have to have λ(t) = 0 for all t [0, 2τ]. Supposing that to be the case, choose δ (0,ε/2) and, in the light of the definition∈ of λ(τ), an ∈ elementy ˆ∗ with σ(ˆy∗,τ) <δ. The nonnegativity and concavity of σ(ˆy∗, )on[0, 2τ] · imply then that σ(ˆy∗,t) (δ/τ)t when τ t 2τ and σ(ˆy∗,t) 2δ (δ/τ)t when ≤ ≤ ≤ ≤ − 0 t τ. But that gives us σ(ˆy∗,t) 2δ<εfor all t [0, 2τ], in contradiction to the≤ property≤ of ε in (4.13). Therefore,≤ λ(τ) > 0, as claimed.∈ We have σ(y∗,τ) λ(τ)when y∗ = 1, and hence by positive homogeneity ≥ k k σ(y∗,τ) λ(τ) y∗ for all y∗ Y ∗. In this inequality, σ( ,τ) is the support function≥ of the convexk k set C(τ), or∈ equivalently of cl C(τ), whereas· λ(τ) is the k·k support function of λ(τ)BY . It follows therefore that cl C(τ) λ(τ)BY ,sothatat least 0 int cl C(τ). ⊃ We invoke∈ next a result of Robinson [22, Lemma 1.1]: If D is a closed convex subset of X Y with projections DX and DY on X and Y ,andifDX is bounded, × then int cl DY =intDY .IntakingD = (x, t, y) gph F¯ x BX ,t= τ ,we ∈ ∈ have DX BX τ and DY = C(τ). So this allows us to pass from the knowledge ⊂ ×{ } that 0 int cl C(τ)tohaving0 int C(τ), as desired. We∈ have established (4.8), and∈ turn now to (4.9). The first thing to observe is that every G¯ L(X IR, Y ) can be identified with a pair (G, g) L(X, Y ) Y under the formula∈ G¯(×x, t)=G(x) tg. Moreover, under this identification,∈ we× get G¯ equal to the expression in (4.1),− due to the choice of norm in (4.7). The next kthingk to observe is that 1 t(F + G)(t− x) t(¯y + g)ift>0, ¯ ¯ − (F + G)(x, t)=(F + G)∞(x)ift =0, if t<0, ∅ so that F¯ +G¯ gives the homogenization of the perturbed system (F +G)(x) y¯+g. Therefore, on the basis of what has so far been proved, we have 3 y¯ + g int rge(F + G) F¯ + G¯ surjective. ∈ ⇔ Hence, through Lemma 4.4, the distance to infeasibility for the system F (x) y¯ is the infimum of G¯ over all G¯ L(X IR, Y ) such that F¯ + G¯ is not surjective.3 k k ∈ × Theorem 2.9 then furnishes the conclusion we wanted in (4.9). THE RADIUS OF METRIC REGULARITY 511 Theorem 4.8 (elaborated distance formula). Let F : X ⇒ Y have a closed, convex graph and let y¯ rge F . Define the convex, positively homogeneous function ∈ h : X∗ Y ∗ ( , ] by × → −∞ ∞ h(x∗,y∗)=sup x, x∗ y,y∗ y F (x) . x,y h i−h i ∈ Then, for the system F (x) y¯, (4.14) 3 distance to infeasibility =infmax x∗ ,h(x∗,y∗)+ y,y¯ ∗ . y∗ =1,x∗ k k h i k k n o Proof. By Theorem 4.7, the distance to infeasibility is 1/ reg F¯(0, 0 0). On the other ¯ ¯ + 1 + ¯ + | hand, reg F (0, 0 0) = (F ∗ )− for the adjoint mapping F ∗ : Y ∗ ⇒ X∗ IR;cf. | k k ¯ + × Proposition 2.7. By definition, (x∗,s) F ∗ (y∗) if and only if (x∗,s, y∗) belongs ¯ ∈ ¯ − to the polar cone (gph F )∗. Because gph F is the closed convex cone generated by (x, 1,y y¯) (x, y) gph F , this condition is the same as − ∈ s + x, x∗ y y,y¯ ∗ 0 for all (x, y) gph F, h i−h − i≤ ∈ and can be expressed as s + h(x∗,y∗)+ y,y¯ ∗ 0. Hence h i≤ + 1 + (4.15) (F ∗ )− =sup y∗ (x∗,s) 1,s+ h(x∗,y∗)+ y,y¯ ∗ 0 , k k k k k k≤ h i≤ n o where the norm (x∗,s) on X∗ IR dual to the one in (4.7) is (x∗,s) = k k × k k max x∗ , s . The distance to infeasibility, being the reciprocal of the quan- tity ink (4.15),k | | can be expressed therefore (through the positive homogeneity of h) as (4.16) inf max x∗ , s s + h(x∗,y∗)+ y,y¯ ∗ 0 . y =1,x,s k k | | h i≤ k ∗k ∗ n o (In converting from (4.15) to an infimum restricted to y∗ = 1 in (4.16), we need k k to be cautious about the possibility that there might be no elements (x∗,s,y∗) ¯ + 1 ∈ gph(F ∗ )− with y∗ = 0, in which case the infimum in (4.16) is . Is it correct then that the expression6 in (4.15) is 0? Yes.) Observe next that,∞ in the infimum in (4.16), s will be taken to be as near to 0 as possible while maintaining s − ≥ h(x∗,y∗)+ y,y¯ ∗ .Thus,s will be the max of 0 and h(x∗,y∗)+ y,y¯ ∗ ,and max x , sh willi be the max| | of these two quantities and x —but thenh thei 0 is superfluous,k k | | and we end up with (4.16) equaling the expressionk k on the right side of (4.14). Corollary 4.9 (homogeneous systems). Let F : X ⇒ Y be sublinear with closed graph and let y¯ rge F . Then, for the system F (x) y¯, ∈ 3 + distance to infeasibility =infmax d 0,F∗ (y∗) , y,y¯ ∗ . y =1 h i k ∗k n o Proof. In this case, the function h in Theorem 4.8 has h(x∗,y∗)=0whenx∗ + ∈ F ∗ (y∗), but h(x∗,y∗)= otherwise. ∞ Example 4.10 (linear-conic case). For any constraint system of type x 0, ≥C A(x) K y¯, with respect to a continuous linear mapping A : X Y and closed, convex≤ cones C X and K Y , → ⊂ ⊂ + distance to infeasibility = inf max d(A∗(y∗),C ), y,y¯ ∗ . + y∗ K , y∗ =1 h i ∈ k k n o 512 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR Detail. This specializes to the mapping F in Example 4.2, using the formula for + + + F ∗ furnished there. We have d(0,A∗(y∗) C )=d(A∗(y∗),C ). − 5. Application to Variational Inequalities In this section, we let Y = X∗ and consider a continuous mapping f : X X∗ along with a nonempty, closed, convex set C X.Wetake → ⊂ normal cone to C at x if x C, (5.1) F (x):=f(x)+NC(x), where NC(x)= ∈ if x/C. ( ∅ ∈ The normal cone is the usual one of convex analysis, so that 1 (5.2)x ¯ F − (¯y) x¯ C and x x,¯ f(¯x) y¯ 0 for all x C. ∈ ⇔ ∈ h − − i≥ ∈ In this case, therefore, solving F (x) y¯ amounts to solving the variational inequal- ity for C and f with “forcing term”3y ¯. The concepts and results in Sections 1–3 can be applied in this framework of 1 variational inequalities to gain information on how the solution set F − (¯y) behaves with respect to perturbations iny ¯, or perturbations that replace F by F +G,which amount to replacing f by f + G.NotethatF is positively homogeneous when f is positively homogeneous and C is a cone, but F is sublinear only when f is linear and C is a subspace of X. If actually C = X, the variational inequality turns into the equation f(x)=¯y and we are back in the context of the Lusternik-Graves theorem, where the derivative mapping Df(¯x):X X∗ enters. Our aim now is to develop results about metric → regularity for more general C in which Df(¯x) and its adjoint Df(¯x)∗ likewise have a role. Our technique will be to rely on coderivatives, utilizing the Mordukhovich criterion in Theorem 3.1. So it will be necessary to restrict our analysis to finite- dimensional X. Coderivatives of the normal cone mapping NC : x NC(x) will be needed. 7→ Clearly, when (¯x, y¯) gph F we havey ¯ f(¯x) NC(¯x) and can work with the ∈ − ∈ coderivative mapping D∗NC(¯x y¯ f(¯x)) : X X∗. | − ⇒ Theorem 5.1 (regularity radius for variational inequalities). Suppose dim X< ,andletf be continuously differentiable. Then, for the mapping F in (5.1) and ∞any (¯x, y¯) gph F , ∈ (5.3) rad F (¯x y¯)= inf d Df(¯x)∗(u),D∗NC(¯x v¯)(u) , where v¯ =¯y f(¯x). | u =1 − | − k k Proof. Because F = f + NC,wehaveD∗F (¯x y¯)=Df(¯x)∗ + D∗NC (¯x v¯)for v¯ =¯y f(¯x) through the coderivative calculus in| [25, 10.43(b)]. Hence by Theorem| − 1 + 3.1, reg F (¯x y¯)equals (Df(¯x)∗ + D∗NC(¯x v¯))− . Proposition 2.5 then gives us | k | k 1 reg F (¯x y¯)− =infd 0,Df(¯x)∗(u)+D∗NC(¯x v¯)(u) . | u =1 | k k Since d 0,Df(¯x)∗(u)+D∗NC (¯x v¯)(u) = d Df(¯x)∗(u),D∗NC (¯x v¯)(u) and rad F (¯x y¯)=1/ reg F (¯x y¯) by Theorem| 1.5, we− get (5.3). | | | Corollary 5.2 (effect of first-order approximations). Under the assumptions in the theorem about F = f + NC, one has rad F (¯x y¯)=radΦ(¯x y¯) for Φ=ϕ + NC | | THE RADIUS OF METRIC REGULARITY 513 whenever ϕ : IRn IRn is continuously differentiable with ϕ(¯x)=f(¯x) and DΦ(¯x)=Df(¯x). In→ particular, this holds for the linearization ϕ(x)=f(¯x)+ Df(¯x)(x x¯). − Proof. The radius formula in the theorem depends on f only through Df(¯x). Especially worthy of attention here is a connection with the following property, which we define in general although planning for now to use it only for F of the type in (5.1). Definition 5.3 (strong metric regularity). A mapping F : X ⇒ Y will be called strongly metrically regular atx ¯ fory ¯,where(¯x, y¯) gph F , if it is metrically regular ∈ and neighborhoods X0 ofx ¯ and Y0 ofy ¯ exist such that for each y Y0 there is only 1 ∈ one x X0 in F − (y). ∈ Through the characterization in (1.4) of metric regularity by way of the Aubin 1 property, this means equivalently that F − has a single-valued localization around (¯y,x¯) that is Lipschitz continuous. (To say that a mapping has a single-valued localization around a point in its graph is to say that the intersection of the graph with some neighborhood of the point is the graph of a single-valued mapping defined on a neighborhood of the domain-projection of that point.) The next theorem underscores the importance of this concept for variational inequalities (5.4) g(w, x)+NC (x) y 3 having not only y but possibly an additional element w as parameters. The proof re- lies on results about polyhedral variational inequalities in the Dontchev-Rockafellar paper [8]. Theorem 5.4 (strong metric regularity from polyhedral convexity). Let dim X< ,andletF be of the form (5.1) with f continuously differentiable and C polyhe- dral.∞ Then at any (¯x, y¯) gph F , metric regularity implies strong metric regularity. Moreover, rad F (¯x y¯) is∈ in this case the largest radius r>0 such that for any con- | tinuously differentiable mapping g : IRd IRn IRn (any d)andanyw¯ IRd satisfying × → ∈ (5.5) g(¯w, x¯)=f(¯x), Dxg(¯w, x¯) Df(¯x) (5.6) S :(y,w) x g(w, x)+NC(x) y , 7→ 3 has a single-valued localization at (¯y,w, ¯ x¯) that is Lipschitz continuous. Proof. For g : IRd IRn IRn continuously differentiable andw ¯ IRd,consider the partial linearization× → ∈ (5.7) fg(x)=g(¯w, x¯)+Dxg(¯w, x¯)(x x¯). − By [8, Theorems 1 and 3], the solution mapping S in (5.6) has a single-valued localization at (¯y,w,¯ x¯) that is Lipschitz continuous if and only if the mapping y x fg(x)+NC (x) y has the Aubin property aty ¯ forx ¯,orinotherwords, 7→ 3 Fg = fg + NC is metrically regular atx ¯ fory ¯.HereFg = F + Gg for Gg =[fg f] − and lip G g(¯x)= Dxg(¯w, x¯) Df(¯x) . We know from Theorem 1.5 that rad F (¯x y¯) is the largest radiusk r>0 such− thatk for any G : X Y with lip G(¯x) 1 + (5.10) ρ (A)=1/ A− , K k K k THE RADIUS OF METRIC REGULARITY 515 where AK is the positively homogeneous mapping defined on X by P (A(u)) if u K, (5.11) A (u)= K ∈ K if u/K. (∅ ∈ In particular, ρK (A) > 0 if and only if AK is nonsingular. Proof. It is enough to observe that d(w, K∗)= P (w) in this Euclidean setting. k K k The formula then follows from Proposition 2.5 as applied to AK , with reference also to Definition 2.2. Corollary 5.8 (subspace restrictions). In the case of the proposition where K is a subspace of X, so that the mapping AK : K K is linear, → 1 1/ AK− if AK is nonsingular, (5.12) ρK (A)=radAK = k k ( 0 if AK is singular. Example 5.9 (radius in the nondegenerate case). Suppose in Theorem 5.6 that X is Euclidean and the variational inequality F (x) y¯ for F (x)=f(x)+NC(x) 3 is nondegenerate at its solutionx ¯, in the sense that the critical cone K(¯x, v¯)isa subspace. Then (5.13) rad F (¯x y¯)=radAK for A = Df(¯x)∗ and K = K(¯x, v¯). | Detail. In this case, the collection in Proposition 5.5 has only one element, namely K = K(¯x, v¯). We specialize (5.9)K to this case and apply the formula in Corollary 5.8. The nondegenerate case in Example 5.9 is generic for polyhedral variational inequalities. All the complications in (5.9) not covered by the simple formula (5.13) are thus associated with various types of degeneracy. As a more specific illustration of how the formula in Theorem 5.6 might be used to calculate the radius of metric regularity, consider the nonlinear programming problem =0 fori [1,r], (5.14) minimize ψ0(z) subject to ψi(z) ∈ 0fori [r +1,m], (≤ ∈ n 2 where ψi : IR IR, i =0, 1,...,m,are functions. In terms of the Lagrangian function → C m L(z,λ)=ψ0(z)+ λiψi(z), i=1 X the first-order (Karush-Kuhn-Tucker) optimality condition for the problem (5.14) at a pointz ¯ concerns the existence of a multiplier vector λ¯ satisfying r m r zL(¯z,λ¯)=0, λL(¯z,λ¯) NΛ(λ¯), where Λ = IR IR − . ∇ ∇ ∈ × + This gradient condition is the variational inequality (5.1) with respect to x =(z,λ) for n f(x)=( xL(z,λ), λL(z,λ)),C= IR Λ, x¯ =(¯z,λ¯), y¯ =(0, 0). ∇ −∇ × 516 A. L. DONTCHEV, A. S. LEWIS, AND R. T. ROCKAFELLAR To compute the radius of the metric regularity of F := f +NC atx ¯ fory ¯,weappeal to (5.9). The cone collection has been described in [8, Theorem 5] as follows. Define the index sets K I1 = i [r +1,m] ψi(¯z)=0, λ¯i > 0 1, ,r , { ∈ | }∪{ ··· } I2 = i [r +1,m] ψi(¯z)=0, λ¯i =0 , { ∈ | } I3 = i [r +1,m] ψi(¯z) < 0 . { ∈ | } Then every cone K in the collection is defined by a partition of 1, 2, ,m K I { ··· } into index sets I10 ,I20 ,I30 with the property I1 I10 I1 I2 and I3 I30 I2 I3 in the following way: ⊂ ⊂ ∪ ⊂ ⊂ ∪ z0 free, λi0 free for i I10 , (z0,λ0) K ∈ ∈ ⇐⇒ λ0 0fori I0 , i ≥ ∈ 2 λ0 =0 fori I0 . i ∈ 3 Then z00 =0, λi00 =0 fori I10 , (z00,λ00) K∗ ∈ ∈ ⇐⇒ λ00 0fori I0 , i ≤ ∈ 2 λ00 free for i I0 . i ∈ 3 Using this particular form of the coneK and its polar, and taking the norm to be the canonical Euclidean norm, we see that for a given x0 =(z0,λ0) K asimple rearrangement gives ∈ 2 2 2 d(Df(¯x)∗(x0),K∗) = L(¯z,λ¯)z0 λ0 ψi(¯z) k∇zz − i∇ k i I I ∈X10 ∪ 20 2 2 + ψi(¯z),z0 + max 0, ψi(¯z),z0 . h∇ i { h∇ i} i I i I X∈ 10 X∈ 20 This expression can in principle be utilized in (5.9). 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Ursescu, Multifunctions with closed convex graph, Czechoslovak Mathematical Journal 25 (1975), 438–441. MR 52:8869 Mathematical Reviews, American Mathematical Society, Ann Arbor, Michigan 48107- 8604 E-mail address: [email protected] Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 E-mail address: [email protected] URL: http://www.math.sfu.ca/ aslewis ∼ Department of Mathematics, University of Washington, Seattle, Washington 98195 E-mail address: [email protected]